DOI: Emergency Medicine doctor. However this does not constitute formal medical advice.
Obviously I don't know a huge amount about the injury. Having said that if they have let you go home without any physical support (corset and the like) or plan for surgery, it would suggest that your surgeon is happy that this is a stable, relatively minor injury which will heal itself. If there is a hint of instability or potential of the fracture position getting worse, they will keep you in hospital, on your back and look to intervene sooner rather than later.
Flying per se would not worsen anything in a minor injury in the spine such as this. (The pressure changes experienced in flight are a concern only in limb fractures with casts on, where the fracture site can swell.)
Given this, my overwhelming intuition would say the chances of any problems being encountered are exceptionally small. Your flight is also over a week away - you will have a good idea of which way your recovery is going by then. Particularly rough turbulence could 'jar' the back and worsen the pain, but then so could a sudden stop in a car on the way to your hospital appointment.
Obviously, take the advice of the surgeon looking after you and the BA medical site.